Fluid Space Theory Offers a Solution to the Galaxy Rotation Problem

By John S. Huenefeld

huesoft@frontier.com

ABSTRACT

It is the purpose of this paper to offer an alternative to dark matter for resolving the

difference between observed rotation rates of galaxies and the rotation rates predicted by

gravitational theory. This shall be accomplished by recognizing an existing property of

ordinary matter that has, until now, been overlooked. The theory presented in this paper

is not a type of MOND (modified Newtonian dynamics) but instead a fully consistent

physical theory which will have implications far beyond the problem addressed in this

paper. It is called Fluid Space Theory. There is minimal need for references, as the math

presented is based on applications of special relativity, calculus, and differential equations

which may easily be found in any number of math and physics text books. In summary,

this paper proposes a formulation of gravitational fields consistent with general relativity,

plus an undiscovered property of ordinary matter in the form of a previously unrecognized

contraction field, and demonstrates that inclusion of this contraction field predicts rotation

curves in spiral galaxies similar to observations without the need for dark matter.

SECTION 1. THE CONCEPT OF VELOCITY FIELDS

In cosmology, the concept of universal expansion is widely accepted. In this theory,

the fabric of space-time expands, carrying galaxies outward like raisins in a rising loaf of

bread. At any given position inside this space-time an invisible expansion field may be

imagined as spheres of space-time moving outward with a velocity increasing in

proportion to the distance from the central point. This is a widely accepted and easily

understood example of a space-time velocity field.

There is another somewhat accepted velocity field. In the description of what is

happening at the event horizon of a black hole, the waterfall analogy is commonly used.

In this analogy, space-time is said to be falling into the black hole at the speed of light.

Like a swimmer upstream of a waterfall, anything caught in this flow will be swept down

into the black hole never to return. This will happen even while swimming at top speed

(which in this case is the speed of light). Once again, an accepted concept of a space-time

velocity field.

There is, however, a problem with the waterfall analogy which is subsequently

ignored. If space-time is flowing into the black hole at the speed of light at the event

horizon, it must also be flowing into a concentric sphere imagined just outside of the event

horizon at a velocity just under the speed of light. In fact this may be imagined to go on

through increasingly larger spheres at lower velocities on out to infinity. Also, if this is

true of black holes, why would it not be true for neutron stars, or normal stars, or large

planets. Gravity is gravity whether generated by a black hole or any other massive body.

If the waterfall analogy applies to black holes it must also apply to stars and planets. It

may then logically follow that all objects with the property of mass must be surrounded by

an inward velocity field.

This is a problem, and a line of thinking that physics professors have been steering

students away from since the time of Einstein. I ask the reader to indulge me and follow

this line of reasoning, as I characterize these inward velocity fields. The first step is to

establish the concept of space-time flux.

Let us begin with flat space-time known in GR as Minkowski space. In the tradition

of Albert Einstein's thought experiments, let us travel in mind to a region far from any

massive bodies, where there are no energetic fields, where parallel lines never meet, and

an object left alone will travel forever in a straight line at a constant velocity. Let us

imagine a glass box measuring several meters on a side in this space. Inside the box are

several objects and a human observer. A human observer is posted outside the box as

well. At present, all these items float weightless and motionless (see fig 1A).

Figure 1.

Inertial and accelerated systems.

While weightless, these objects could be traveling through space-time at any velocity

from zero to c (the speed of light). They would have no way to tell what that velocity is,

but whatever it is, they can tell it is not changing. We could then say that the velocity

field, or space-time flux, inside the box is constant. If we discard any unknown

background velocity, using the cross sectional area of the box we can compute the relative

flow of space-time through the box. We don‘t know an absolute flux, but a relative flux can

be described by equation (1). The volume flux will have units of meters cubed per second,

m3s-1. All further references to space-time flux may be considered as “relative flux.”

(1)

In Figure 1B, a force has been applied to the top of the box and now the observer

inside the box experiences an increasing space-time flux. Space is flowing through the box

at a constantly increasing rate and she experiences gravity. The change in space-time flux

with time can be expressed by equation (2). The change in space-time flux will have units

of m3 s-2, and these are the units that provide the experience of gravity.

(2)

Understanding that acceleration is the time derivative of velocity this equation may be

rewritten as equation (3).

(3)

In figure 1B, a force was required in order to create a field of changing velocity. The

equation can be expressed in terms of that force and the mass of the glass box and its

contents as in equation (4).

(4)

Now we shall return to the problem of a spherical inflow velocity field around a planet.

Imagine now that instead of being pulled trough space by a force, the glass box is sitting

on the surface of a planet. The astronaut inside the box will feel the same sensation of

gravity. There is a name for this, it is called the equivalence principle, named by Einstein

himself. The equivalence principle states that there is no distinction between the gravity

felt while standing on the surface of a gravitating body or while being pulled through

space in a box. This means that on the planet’s surface, the flux of space time must also be

constantly changing with time. We may then replace the term for area A in equation (4)

with the surface area of the planet, which has radius r to arrive at equation (5).

(5)

While the box is sitting on the surface of the planet, the force of gravity felt by the

observer inside remains constant. This is also true for the box being pulled through space,

as long as the force remains constant. Looking at the term V double dot, we see it has

units of m3s-2 This is very similar to the units of the gravitational constant G which has

units m3kg-1s-2. If V double dot is assumed to be proportional to the mass M of the planet,

we may replace the change in space-time flux term with a convenient constant, 4 pi G,

which is applied in proportion to the mass M of the planet and we get the very familiar

equation (6).

(6)

In this way, Newton’s equation for gravity may be derived on the basis of a space-time

inflow velocity field. Of course, there is a problem with this. If space-time is flowing into

the planet from all sides, where is it going? The planet should quickly fill up with space-

time and the flow will have to stop. Also the flow velocity at the surface will have to be

constantly increasing with time to create gravity, making the situation even worse. The

amount of space-time volume passing through any imagined sphere at diameter D must be

accounted for, as well as the second order term for ever increasing velocity, and this seems

impossible.

At this point, most people have thrown up their hands and walked away from this line

of thinking, but not me. There is an answer, and it is precisely by accounting for this “lost

volume” of space-time, that the cause of galaxy rotations faster than predicted by

Newton’s gravitation equation (dark matter) can be explained.

SECTION 2. THE APPLICATION OF RELATIVITY TO VELOCITY FIELDS

Understanding how inflow volumes may be accounted for came as a “happy idea.”

Time and space, said Einstein, are not the rigid and inflexible things that Newton thought

them to be. Time does not pass at the same rate in all references frames, nor are distances

the same. He gave us the equations below that can be used to predict the rate of time and

the lengths of objects based on their relative velocities. These equations of special

relativity known as Lorentz transformations, equations (7a) and (7b), will be valuable in

applying relativity theory to spatial inflows.

(7a)

(7b)

In these equations, variables marked prime are those in the moving reference frame

while the unmarked variables are in the rest reference frame of the observer.

The next thing to consider is that objects that we perceive as solid are actually, almost

completely made up of empty space, it is only the fields surrounding very tiny objects at

the heart of matter that makes them seem solid. So when we use the equations of special

relativity to compute the length of an object traveling near the speed of light, we are

actually computing the changes in the space that object occupies. Specifically, the

coordinate axis in a moving reference frame, aligned in the direction of motion, will appear

to be compressed to an observer in another reference frame. Any object placed into that

moving reference frame will also appear compressed along that axis.

Now let’s turn back to the notion of space-time falling through the surface of an

imaginary planet. Imagine a small cube of space-time as it moves toward the surface of

the planet. As the block of space-time falls inward, it moves faster and faster. The faster

it moves, the shorter it becomes in the direction of motion (the radial direction), and its

internal volume decreases. If it can fall fast enough to reach the speed of light, its volume

will vanish entirely. In this context, the notion of space-time flowing into matter is not so

absurd after all. The inflow of space-time vanishes as it becomes compressed with

increasing velocity.

Before developing the mathematics of an inflow field I wish to establish a couple of

conventions. First radial vectors are defined as positive outward and negative inward

from the central point of a flow field. Second, I will use the diameter rather than the

radius of a sphere in the equations. With spatial compression in the radial direction, the

term radius can lead to some confusion because distances in the radial direction are not

constant. The diameter however remains a consistent measure. Finally when applying

relativity to the flow field, one must consider both the view of an observer greatly

removed, on the outside of the flow, and the view of an element of space-time traveling

within the flow field. These two views can become very different.

Starting from Newton’s equation and substituting diameter for radius (D = 2r) and

using Newton’s force relation (F=ma) to express inward acceleration rather than the force

of gravity, and we set acceleration equal to the time rate change of velocity we get

equation (8).

(8)

Next by the chain rule we look for the velocity change with respect to the diameter,

realizing that the time change in diameter of a falling shell is equal to 2v. We may now

solve for the change in velocity with respect to the diameter dv/dD as show in equations (9)

through (12).

(9)

Therefore.

(10)

Integrating both sides we get.

(11)

(12)

We recognize equation (12) as the Newtonian formula for escape velocity from a

gravitating body. This is the unique inflow velocity profile that results in an inverse

distance squared acceleration profile. In this case, we are not considering a body falling

through Newtonian space but the fabric of space-time itself, falling toward a central point.

This is commonly referred to as sink flow. At any diameter in a gravitational field, space-

time falls inward at escape velocity.

If space-time is considered an incompressible fluid, such an inflow would be

impossible. It is easy to calculate that the flux of space-time changes as a function of

diameter D. However, if space-time is allowed to compress as it flows inward, there is a

chance to account for the change in flow volume with diameter. This may be done by

adjusting the radial length of the flow for the effects of relativity. The Lorentz

transformations of Special Relativity may be applied to correct for both spatial

compression and time dilation as shown in equations (13) and (14).

(13)

(14)

Substituting the value of v from equation (12) into equation (14) we get the relativity

corrected equation for inflow velocity as a function of the diameter as observed from a

reference frame outside the flow field, equation (15).

(15)

Equation (15) represents the velocity field as a function of diameter that will be

observed around any object that has the property of mass according to Fluid Space Theory.

Figure 2. is a plot of this function. In this graph v is the Newtonian form of the velocity

profile and v prime is how the velocity will appear to an observer outside the flow field

after accounting for spatial contraction and time dilation.

Figure 2.

Newtonian and relativistic velocity profiles.

There are a few things to note about this graph. As with all similar figures presented,

there are no scales associated with either axis. The graph has been scaled to illustrate the

shape of the function. We must think of the v curve as what an element within the flow

field will see while the v prime curve represents the view of that element from outside the

flow field. Consistent with cosmological expansion theory, the inflow v may become

superluminal (exceed the speed of light) while the function for v prime falls off to zero at a

short distance before D equals zero. When v exceeds c, the element within the flow will

vanish at some minimum diameter, as far as the outside observer is concerned. That

diameter can be found by solving for D when v prime equals zero as shown in equations

(16) and (17). (This is the same as setting the flow velocity v equal to the speed of light c).

(16)

(17)

We recognize this equation as the Schwarzschild diameter (twice the radius). This is

the diameter at which the inflow comes to a stop from the point of view outside the flow

field. Relativity may be applied to the Newtonian form for acceleration in the flow field

using the same technique to obtain equation (18), the observed acceleration, a prime.

(18)

Figure 3 is a plot of the flow acceleration as a function of the diameter. In this graph,

a is the Newtonian form of the acceleration field and a prime is the acceleration field as it

would appear to an observer outside the flow field.

Figure 3.

Newtonian and relativistic acceleration profiles.

Knowing that the acceleration is inward, it has been plotted on the positive axis in

figure 3 to keep the graphs consistent with convention. We see the acceleration increasing

as we move toward the central point, matching Newtonian gravity. Then, at small

diameters, the a prime curve reverses and slows until it becomes zero at the

Schwarzschild diameter. You must look closely at this graph to see how a prime follows

the Newtonian form but then drops away leaving a sharp peak at 1.5 times the minimum

diameter.

So it may be possible to account for vanishing volume in a velocity field, but how can

the constantly increasing velocity of inflow at a planet’s surface be accounted for? It so

happens that a sphere is actually a form of funnel. It is the most extreme form where the

angle of the vertex is opened all the way to 360 degrees. In a flow down a funnel the

velocity of a fluid will naturally increase toward the narrow end. In this way, an

acceleration field will be superimposed on the velocity field while the flow rate can remain

in a steady state. The imposed acceleration field will create gravity (it actually is gravity).

It is important to note at this point that the superimposed velocity and acceleration

fields described by equations (15) and (18) constitute a gravitational field completely

consistent with General Relativity. The fields are nearly identical to those specified by

Einstein’s field equation. These inflow fields produce gravity, curved space-time, and

gravitational time dilation. The differences between these fields and GR fields are subtle

and that is a subject for another paper.

So now, it has been shown that not only Newton’s equation can be derived from space-

time inflow fields, but a form of General Relativity is arrived at as well. This provides a

solid foundation for moving forward even though space-time is not usually treated this

way.

Figure 4. is a diagram of Fluid Space Theory funnel flow. It is useful for establishing

parameters for describing “lost volume” when setting up the equations of fluid space flows.

In this figure, the flow field outside view and inside view are superimposed.

To move ahead, it will be helpful to think in four dimensions. Four dimensional space-

time, is composed of the three familiar spatial dimensions x, y, z, and an additional

dimension tc which exists on the time axis. According to special relativity theory, for any

contraction on the x axis, there is an equivalent expansion on the tc axis. The tc axis must

also be considered perpendicular to all three spatial axes.

While three dimensional volume is not conserved in spatial inflows, four dimensional

volume is conserved. Four volume is defined simply as the product of the four dimensions

as in equation (19).

(19)

Under a velocity transformation four volume is unchanged as shown in equation (20).

(z=z’ and y=y’).

(20)

While contraction on the x axis might be noticed by the outside observer, expansion on

the tc axis is much harder to detect or comprehend and is generally overlooked.

Application of time dilation is almost exclusively reserved for high energy physics.

In Figure 4, space-time is imagined to be flowing down a funnel, which can also be

thought of as inflow from all directions into a sphere. The straight taper is what the

outside observer will see (assuming flat space) while the curved, hyperbolic funnel

represents what is actually going on for an element inside the flow field (curved space). At

an infinite diameter the curved funnel will be tangent to the straight funnel and the

compression will be zero. At Dmin (v=c), the curved funnel will be tangent to a line

Dmin/2 off the radial axis and the compression will become infinite (volume equal to zero).

Figure 4.

The two views of spatial in flow.

By the time an element in the flow field reaches an arbitrary diameter D, due to

spatial compression, it will have traveled an additional distance l down the curved funnel,

from its point of view, than what is observed from outside the flow field. The shaded

region lA represents the volume of space that has been compressed at any diameter D.

The flow or flux of space-time passing through a sphere of diameter D has been

established in equation (1), and is repeated here for convenience.

(1)

Substituting the value of v from equation (12) and the area for a sphere we get

equation (21).

(21)

This is the “true” volume of spatial flow as seen by elements embedded within the flow

field. The flux of space-time as observed from outside the flow field, accounting first for

spatial compression is shown in equation (22).

(22)

The amount of spatial compression taking place at any diameter D will be the

difference between the inside view and the outside view, that is the difference between the

uncompressed flow going down the hyperbolic funnel and the compressed flow observed

going down the straight funnel. This is shown in equation (23).

(23)

We may now apply time dilation to arrive at a complete expression for “lost flux” as a

function of diameter in equation (24).

(24)

Figure 5. is a plot of this function and both spatial compression volume dot and

volume dot prime are shown. What this graph shows is a bit surprising. The compression

rate volume prime starts at zero at Dmin and increases parabolically with diameter. The

lost volume curve without accounting for time dilation is similar and may be used at very

large diameters (that is the only reason it is shown).

The meaning of this function is that while the effects of relativity diminish as

diameter increases, because of the rate volumes increase with diameter, there is a

significant and ever increasing spatial effect. Any dark matter physicist should recognize

the shape of this curve. It is the shape of the gravitation curve for dark matter required to

flatten a galaxy rotation curve.

Figure 5.

Lost Flux as a function of diameter.

SECTION 3. THE SECOND (MISSING) COMPONENT OF GRAVITY

The concept of space-time flow fields will take some getting used to. This is not the

way physicists are trained to think. It may be time for the reader to ponder the first two

sections of this paper for a while. There are some important things to keep in mind. The

inflow velocity field by itself is invisible and can be superimposed over any other velocity

field without consequence. The inflow velocity field will not sweep objects toward the

center of the flow other than in conjunction with the acceleration field. It is the change

with respect to time in the velocity field that creates the acceleration field which we call

gravity. The acceleration field can be observed, and may also be superimposed over other

accelerations fields to build up an accumulated field which will behave according to the

distribution of masses within it. It is, however, the magnitude of the accumulated velocity

fields that determines the curvature of space-time.

In the background of the Fluid Space Theory gravitational field there is a second field.

The “lost flux” field, which represents a contraction of space-time surrounding all objects

that have the property of mass. This lost flux field constitutes the volume of space-time

which has been compressed by relative velocity and thereby shifted over to the tc axis. As

such it must be considered a separate field, orthogonal and acting independent from the

primary gravitational field. Therefore, the effects of this second field cannot be simply

added to gravity. It must be dealt with separately.

The lost flux field manifests as a contraction around matter in a spherical shell as a

function of diameter according to equation (24). Dividing equation (24). by the area of the

sphere yields what may be called the drift velocity.

(25)

This represents the velocity at which objects in the lost flux field will be swept toward

the center. It has been computed on the basis of the diameter so to find the radial drift

velocity the value must be cut in half to arrive at equation (26).

(26)

This function is plotted in Figure 6.

Figure 6.

This function has a similar form to the acceleration curve in Figure 3 but it is a

velocity curve, first order with time, while the acceleration curve is second order with

time. In order to account for the complete motion of a particle in a gravitational field both

equations (18) and (25) must be applied. Equation (18) will dominate out to very great

distances but eventually the drift velocity becomes equal to the gravitational acceleration

produced per unit time. After that, the drift velocity may become many times greater than

the gravitational acceleration.

The method I have employed to calculate the additional orbital velocity required to

overcome the inward drift, is to assume the drift is created by an acceleration which will

produce the same value as the drift velocity over a period of unit time. First we calculate

the acceleration required to produce the drift velocity per unit of time to arrive at equation

(27).

(27)

Next this term is combined with the normal gravitational acceleration to create a scale factor as shown in equation (28).

(28)

The total orbital velocity is then computed using the scaled acceleration as in equation (29).

(29)

To illustrate the long range effect of this newly revealed component of gravity I have

prepared a model of our solar system and a crude model of a galaxy based roughly on the

size of the Milky Way. In these models, normal gravity has been computed according to

equation (18) and the drift velocity has been computed according to equation (26). The

total adjusted orbital velocity is computed by applying the scale factor to the normal

gravitational acceleration. This is quite similar to the dark matter method of computing

additional gravity created by an assumed unseen mass. Mass and acceleration are in

direct proportion in the gravity equation, so the dark matter theorist scales up the mass

while Fluid Space Theory scales up the acceleration. While dark matter theorists must

assume unseen matter, the contraction field of Fluid Space Theory is deduced through

logic and reason.

Figure 7. is a plot of the orbital velocities for the planets in our solar system predicted

when both equations are applied. In this figure, orbital velocity (for circular orbits) in m s-

1 is plotted on the vertical axis while the horizontal axis has no scale, our solar system’s

features are simply listed in order from the inside out. As you can see, the predicted

values match well with observations. The new corrected values add a small and nearly

constant amount to the values predicted by standard gravity. However, this represents an

increasing percentage of the orbital velocity with increasing distance from the sun.

Figure 7.

There may have to be a re-evaluation of the value of the gravitational constant G.

Until now, G has been computed based on the assumption of a single component

gravitational field. In light of the additional contribution of the contraction or drift field,

G may have to be changed slightly from its current value in order to match observations.

This may actually help establish G with greater precision and could be the reason for

variations in the measurement of G carried out by different methods at different

distances. In addition, this may also help predict the orbits of Oort cloud bodies where the

contraction field contribution becomes more significant.

Figures 8. and 9. are based on the galaxy model. The model was created in an excel

spread sheet by breaking the galaxy into 16 primary zones 1,000 parsecs wide containing

galactic matter with four additional 1000 parsec wide zones containing diminishing

amounts of matter to fade out the galactic rim.

A super massive black hole of 2.6 million solar masses was placed at the center. Each

zone was represented by a concentric ring 1000 parsecs wide located outside the previous

zone. The galactic disk thickness was set to 600 parsecs at the core (central cylinder) with

tapering thickness down to 100 parsecs at the 16,000 parsec outer radius ring. The

remaining four rim rings tapered to 30 parsecs. Masses for each ring were calculated by

multiplying the volume of the ring by an estimated stellar density. The stellar densities

also diminish in magnitude from the core outward. The density in zone 1 was set high to

simulate a galactic bulge with the remaining zones having much lower densities. The

target mass was just over 20 billion solar masses (200 billion if you assume dark matter).

Figure 8.

Stellar orbital velocities in m s-1 predicted by combined fields. The blue line, velocity

from G, is the contribution from gravity alone. The red line, velocity from C (compression),

is the contribution from the compression field.

Figure 9.

Galactic mass distribution in solar masses.

Orbital velocities were calculated at the outside of each zone based on the

accumulated mass of all the zones inside. Because of the crudeness of the model, the plot

jumps up quickly on the left side near the core. A finer spacing of data points near the

core would smooth the curve. However, this model was only intended to test Fluid Space

Theory for the prediction of higher orbital velocities outside the core than predicted by

gravity alone. As you can, see it does that very well, predicting a quite flat total orbital

velocity curve all the way to the galactic rim.

The acceleration scale factors computed for each zone are listed in Table 1.

Zone Scale Factor Zone Scale Factor

1 3.04 11 8.41

2 3.75 12 8.79

3 4.40 13 9.16

4 5.01 14 9.50

5 5.59 15 9.82

6 6.13 16 10.13

7 6.63 17 10.42

8 7.11 18 10.70

9 7.57 19 10.97

10 8.00 20 11.23

Table 1.

From this simple model, acceleration scale factors reached values more than 10 times

that of gravity acting alone. The long range nature of the contraction field is also revealed

with scale factors climbing slowly from the galactic core and continuing to climb all the

way to the galactic rim, even while galactic mass content was tapering off. This

completely replicates the results of a dark matter halo, without the need to have any dark

matter at all.

SECTION 4. CONCLUSIONS

Let me start by saying that I did not create Fluid Space Theory to solve the galaxy

rotation problem. This is a recent and unexpected discovery, and a new application of a

well considered and documented (if not widely accepted) theory. I’ve been working on it

since the late 1990’s when the “happy idea” of how inflow volume is lost occurred to me.

For many years, as I explored the theory, all I seemed to have was an easier way to

explain and understand General Relativity. Each new conclusion and prediction I made

was already covered by GR. It was like finding little road signs along the way saying

“Albert Einstein was here.”

Extraordinary claims require extraordinary proof. I believe this result is

extraordinary and worthy of further consideration. It is significant because it is based on

a whole and self contained theory of the universe and it is not a patch or band-aid slapped

onto an existing theory in order to make it match observations. And at last, I have found a

place along the road without any sign from Albert Einstein.

Fluid Space Theory replicates many features of General Relativity, but it also departs

from GR in several important ways. In the form presented in this paper, Fluid Space

Theory is not fully generalized and I hope more refinements and investigation can produce

better results and a fully generalized theory may be published at a future date. I look

forward to working with physicists interested in exploring this line of reasoning.

The galactic contraction fields predicted will be in the form of a spherical halo and will

cause gravitational lensing exactly as do the dark matter halos they replace. In this case,

the contraction fields are predicted on the basis of the properties of known matter and

generally accepted properties of space-time. These fields will require a name, and I

humbly submit the term, Mannfields in memory of my mother and uncle who left this

world in 2014.

How this theory affects cosmology is not clear to me at the time of writing this paper.

Galactic contraction fields could certainly offset the rate of overall universal expansion

and may increase the predicted age of the universe. These fields, while diminishing in

magnitude outside a galaxy, will still continue to outstrip the effects of gravity and may

also change the dynamics of galaxy clusters.

The very long range influence of these fields has only been marginally investigated.

Computations of the vacuum energy curve for the Milky Way galaxy model above show

negative values just outside the galactic core which remain negative out to the rim and

beyond. This is consistent with a contracting space-time within galaxies. Interestingly at

about 90 parsecs, the vacuum energy turns positive again and remains so on outward.

This hints that Fluid Space Theory may also hold the key to explaining dark energy as an

effect of normal gravity as well.

Perhaps the most surprising thing about this prediction is that it is the result of

relativity acting at very large distances, and very low velocity. Relativity is not supposed

to be significant in these areas, so such a finding is quite unexpected. Unexpected

discoveries are sometimes the most satisfying and this actually gives me greater hope that

my proposal will prove out.

I have some small reservations about the acceleration scaling method applied. It

works well enough on the galactic scale, but I believe a better, more exact solution to

calculate orbital velocities in a contraction field may be found. I would not be surprised if

this has already been done by a mathematician working on first order attraction fields. I

will continue investigating an exact solution for the drift field and provide updates if

successful. Investigation of the action of the contraction field with a higher fidelity

galactic model, including dust, spiral arms, and actual mass distributions, is currently

beyond my means. I would gladly join such an effort if presented with the opportunity.

In summary, this paper proposes a new formulation of gravitational fields consistent

with general relativity, an undiscovered property of ordinary matter in the form of a

previously overlooked contraction field (the Mannfield), and demonstrates that this

contraction field predicts rotation curves in galaxies similar to observations without the

need for dark matter.

© 2014, by John S. Huenefeld, Wednesday, December 24, 2014

All Rights Reserved. This paper may be shared for review and comment purposes only.